Two new investigations into test-tampering by teachers and administrators in city schools are under way, The Post has learned.

The state Education Department and city Board of Education are jointly investigating allegations that administrators at several high schools changed or discarded failing scores on state exams to improve their overall statistics.

In addition, Special Schools Investigator Ed Stancik is probing more than a dozen new allegations that surfaced after he issued a sweeping report in December charging test-tampering in 32 schools citywide.

Stancik’s investigation includes three complaints stemming from last week’s fourth-grade reading and writing tests, said board spokeswoman Pam McDonnell.

The state probe covers six schools, including accusations that top administrators at William Grady HS in Brooklyn have for years tossed in the trash failing score sheets on math and science Regents exams.

Whistle-blowers say administrators claimed the abysmal scores would ruin kids’ records. Students who fail Regents exams can take them again.

But eliminating the failures also improved the school’s overall passing rate, they said.

“It’s just a game to make us look good on paper. Whether our kids learn is irrelevant,” one teacher said.

Grady Principal Anthony Valenti did not return calls yesterday.

Another case involves William Taft HS in The Bronx, where the failing scores of 27 students on a math Regents exam last year were allegedly changed to passing. Taft was recently removed from the state’s list of failing schools for showing improvements.

Stancik’s December report accused 52 teachers and administrators of coaching students on standardized tests, or giving them answers.

It also accused the board of failing to investigate and punish cheaters — and not telling parents about tainted scores.